Theorem REF  shows that all limiting points of the iteration sequence are stationary, while it does not suggest that \(({x}_t, {\lambda }_t)\)  always converges to a stationary point or a local solution.
This is similar to deterministic SQP guarantees [1]}.
The convergence of \(({x}_t, {\lambda }_t)\)  is equivalent to the convergence of \(\nabla \mathcal {L}_t\)  for problems where the objective is strongly convex and constraints are affine.
For nonlinear problems, if \(\mathcal {X}\times \Lambda \)  has a single stationary point inside, then \(({x}_t, {\lambda }_t)\)  also converges to that point.
Furthermore, if that point is a strict local solution (by chance), then \(({x}_t, {\lambda }_t)\)  converges to that strict local solution.
To facilitate our analysis on the convergence rate of StoSQP, we assume in Assumption REF  that \(({x}_t, {\lambda }_t)\rightarrow ({x}^\star , {\lambda }^\star )\)  for a strict local solution \(({x}^\star , {\lambda }^\star )\) , whose Hessian \(K^\star = \nabla ^2\mathcal {L}^\star \)  is invertible (as implied by (i) \(\nabla _{{x}}^2\mathcal {L}^\star \)  is positive definite in \(\lbrace {x}: G^\star {x}= {0}\rbrace \) ; (ii) \(G^\star \)  has full row rank).
Note that \(({x}_t, {\lambda }_t)\rightarrow ({x}^\star ,{\lambda }^\star )\)  is equivalent to assuming that \(({x}_t, {\lambda }_t)\)  jumps into a small neighborhood of \(({x}^\star ,{\lambda }^\star )\)  for a sufficiently large \(t\) .
This is because \(\Vert ({x}_{t+1}-{x}_t, {\lambda }_{t+1}-{\lambda }_t)\Vert  \vee \Vert \nabla \mathcal {L}_t\Vert \rightarrow 0\)  implies that there exists an attraction neighborhood around \(({x}^\star , {\lambda }^\star )\) —once \(({x}_t, {\lambda }_t)\)  lies in the neighborhood, then all the following iterates stay in the neighborhood.
We emphasize that targeting on an iteration sequence that converges to a strict local solution is a routine setup for convergence rate analysis of deterministic SQP (e.g., [2]}, [3]}, [4]}, [5]}); and it is reasonable for StoSQP as well, given the convergence guarantee in Theorem REF  matches the one of deterministic SQP, up to a a zero-probability event.
